No slip boundary condition fully developed flow

Lets assume at the inlet of a (cylindrical) pipe there is a constant water flow with parabolic velocity profile .

I know that fully developed flow remains unchange when moving along the pipe (flow profile remains the same ) And development of flow in pipe occurs due to viscous forces acting on flow.(simply when adhesion between flow particles in contact with wall and wall is greater than the cohesion between fluid particles themselves, so the relative velocity of the layer of the fluid in contact with the wall with respect to the wall is zero. But preconditions to all this, is having a viscous fluid (and a not completely smooth surface of interior pipe wall)

Let's concentrate on fluid which is water, water is not considered a viscous fluid so there is no force acting on the layer of flow in contact with the wall to slow it down. (I know in reality even in this stiation there is ,but we can ignore it)

A) so due to cohesion, the layers with slower velocity tend to have increased velocity so to satisfy mass conservation the central layers should slow down a bit and we will have a velocity component along radius (from the center to the wall) until the complete conversion of parabolic profile to uniform profile . Is it true?

B) is it acceptable to simply assume that the flow profile remains the same? (Cause it indirectly results in acceptance of no slip boundary condition and fully developed flow)

C)what if instead of flow in a pipe, the domain is a porous media with uniform distribution of pores, can we neglect the radial component of flow? (If the flow is not turbulent)

Water, as almost any other fluid in the world, is viscous. Viscous fluids in the regime where continuum hypothesis holds, needs no-slip, no-penetration boundary conditions at solid walls, i.e.the velocity of the fluid at the wall equals the velocity of the wall itself.

A. It's false. At very low Reynolds numbers, the flow is stable and viscosity tends to develop the Poiseuille parabolic axial velocity profile, with no radial or azimutal velocity components. At high enough Reynolds numbers the flow is not steady anymore, being turbulent. The local direction of the velocity depends on space and time. If you think at the average flow, averaging out all the perturbations around it, the profile tends to a developed turbulent profile after some length from the inlet of a straight pipe. The velocity profile is not parabolic anymore:

• its quite uniform in the central region of the pipe;
• you can define a thin viscous layer close to the wall, with large velocity gradient (the higher the Reynolds number, the thinner the layer qualitatively)
• an intermediate, inertial, region blending the viscous region and the csntral region of the pipe.

B. What is acceptable or not usually depends on the regime of motion, on the geometry of your problem, and on the need for accuracy you have.

As written in point A., after a certain distance from the inlet, you could assume that flow is fully developed and the velocity profile (the average profile, if the regime is turbulent) is the same along a straight pipe with constant section.

• Thanks for your answer. Does the same holds if the pipe is filled with particles (porous media)? About not having velocity radial component
– Tbt
Dec 27, 2022 at 19:14
• I'm not an expert of porous media, but I guess that if porosity is something statistically uniform along the pipe, the axial direction is homogeneous for the problem, so the (average) profile is constant along the pipe. BTW: porous doesn't mean with particles inside; the turbulent flow is not 100% axial, since in the boundary layer there's a small radial component. Dec 27, 2022 at 19:24